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  • Find the value of <img alt="\int_{0}^{1} x^{2}\left(1+3 x^{3}\right)^{5} d x" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cint_%7B0%7D%5E%7B1%7D%20x%5E%7B2%7D%5Cleft%281&plus;3%

Find the value of \int_{0}^{1} x^{2}\left(1+3 x^{3}\right)^{5} d x

Option: 1

\frac{457}{3}


Option: 2

\frac{457}{6}


Option: 3

\frac{455}{6}


Option: 4

\frac{455}{3}


Answers (1)

best_answer

Let u=1+3 x^{3}

\begin{array}{l}{d u=9 x^{2} d x}\end{array}

\frac{1}{9} d u=x^{2} d x

To adjust the limits of integration:

When x=0, u=1+3(0)=1

and when x=1, u=1+3(1)=4 

Then

\int_{0}^{1} x^{2}\left(1+3 x^{3}\right)^{5} d x=\frac{1}{9} \int_{1}^{4} u^{5} d u

\frac{1}{9} \int_{1}^{4} u^{5} d u=\left.\left(\frac{1}{9}\right)\left(\frac{u^{6}}{6}\right)\right|_{1} ^{4}=\frac{1}{54}\left[(4)^{6}-(1)^{6}\right]=\frac{455}{6}

Posted by

Sayak

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